It took me a little longer than most to figure out exactly what this fan was doing; but once I finally "got it", I was intrigued, if not a bit skeptical, by the idea. I own a Dyson vacuum cleaner, and while it is not "magic", it is very well engineered; so, I really looked forward to having a bit of time to ponder the claims. Basically what we have in the Dyson fan would appear to be a very conventional fan that is carefully designed to be coupled with its surrounding environment in a way that magnifies the effect one might expect from the otherwise fairly ordinary fan. But it is important to note that this is engineering and not some fundamental departure from the laws of physics. Following is a brief mathematical explanation of how this might be achieved.
As mentioned, the Coanda and the venturi effect lie at the heart of the Dyson design. These effects rely on a high velocity stream to "entrain" or mix with a lower velocity stream in a similar direction. The only real distinguishing factor between the effects used in Dysons fan design and a conventional fan are the efficiency and evenness of the flow.
If we assume that the power in the air stream produced by the fan is:
P = C * A * V^3
Where C is the product of the various constants, A is the swept area of the fan and V is the velocity of the air stream, then it is fairly trivial to re-write this as follows:
Po ~ Pi => C * A1 * V1^3 = C * A2 * V2^3 ==> A1 * V1^3 = A2 * V2^3
That is simply to say, as the area of the wind stream increases, the velocity of the wind stream decreases proportionally. This general notion holds for both a conventional fan and Dysons design; what separates the two models is the efficiency with which this proportionality is maintained. Now, Dysons claim of a 15X increase in the wind stream volume is problematic in that there is no clarification about WHERE this increase is measured. But lets leave that alone for a moment and focus on what a 15X increase in air volume means from a velocity point-of-view. Assuming there are no losses associate with the Dyson process, and that an "increased volume of 15X" really means "15X the Area"
A1 * V1^3 = (15 * A1) * V2^3 ==> 15 = (V1/V2)^3 => V1/V2 = 2.466
That is, V1 would be 2.466 times that of V2. This calculation obviously leaves a lot of real-world factors out, and the assumption that "15X Volume = 15X Area" is dubious at best, but the notion that a 15X increase in volume could be achieved with a fairly small decrease in stream velocity is demonstrated, lending credence to the theory behind Dysons fan.
It is a very interesting concept, and would certainly make an interesting conversation piece, but the price tag is pretty steep if one is simply looking to move a bit of warm air around a room. I would be interested to find out if the air stream produced at a specific distance and speed were in fact achieved at a lower electrical power input than a conventional fan, but I suspect making a meaningful comparison would be extremely difficult.